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MATH 265 Spring 2005 Practice Final from Fall 2004 1. (a) Find parametric equations for the line through (2, −3, 1) parallel to the vector from (2, 0, 0) to (4, 3, 1). (b) Find an equation for the plane through the origin that is perpendicular to the line of part a). 2. (a) Compute the gradient of F (x, y, z) = x2 + ex sin y − z at (0, 0, 0). (b) Find the equation of the tangent plane to the surface F (x, y, z) = 0 at (0, 0, 0). H 3. Evaluate C (2x2 + 1) dx + (xy) dy where C is the circle of radius 2 with center at the origin and counterclockwise orientation. 4. Find a vector in the direction in which f (x, y, z) = x2 y 3 z + 3z 2 increases most rapidly at P = (−2, 1, −1). What is the rate of change in this direction? 5. Set up, but do not evaluate, an integral in cylindrical coordinates for the mass of the solid lying between the paraboloids z = 2 − x2 − y 2 and z = x2 + y 2 if the density function is δ(x, y, z) = z. R 1 R y2 6. Reverse the order of integration in the integral 0 0 f (x, y) dx dy. 7. For the vector field F(x, y) =< 3x2 y 2 + 2y, 2x3 y + 2x > either find a potential function f so that F = ∇f or explain why no such potential function exists. 8. Find and classify the critical points of f (x, y) = x2 + y 2 + x2 y + 4. 9. Given the planes x + y − z = 2 and 3x − 4y + 5z = 6: (a) Find the angle between them. (b) Find symmetric equations for their line of intersection. 10. Let S be the surface z = 4 − x2 − y 2 with z ≥ 0. (a) Compute the surface area of S. (b) Let F =< −y, x, xyz 4 > and nRRthe upward unit normal vector on S. Compute the surface integral S ∇ × F · n dS.